First and foremost, I agree that by limiting yourself to Fender, you could be missing out on a lot of cool options that you at least want to check out before you dismiss. Nevertheless, you really can't go wrong with Fender. I think whether you go for a P-Bass, a J-Bass, or a P/J-combo, which really is a nice compromise of the two, you're gonna ultimately be happy with your choice. And nothing that I say, or that anyone else is going to say is going to do you nearly as much good as just getting out there and beating around on them yourself, getting a feel for it yourself.
That being said, lets talk a little bit about the differences between the two.
The biggest difference in sound between the two is the split coil in the P-Bass is much chunkier sounding, while the single coil jazz pickups blended has a much smoother sound. Other variables play into the sound (I.e. 4 or 5 string, maple, rosewood, or pao ferro fretboard, active or passive, etc) but comparing apples to apples (four string rosewood passive each) the P-Bass will be just a little bit louder than the J with the volume set the same on the amp, and will be thicker sounding. But the J-Bass will be brighter and much much cleaner sounding. It's not so much that it lacks raw low end, indeed, I would call the two about equal in terms of low frequency sounds. But the P-Bass is going to stand out in the mix more (probably a good thing for Punk, probably a bad thing for Folk)
Also, with the P-Bass, you only have one pickup, one volume knob, and one tone knob. That tone knob does provide, certainly, a decent amount of variety, but can not even come close to matching the variety you have in a J-Bass. if you roll it all the way to the bridge pickup, you've got a very bright, very thin, slightly dirty sound that is -fantastic- for soloing (think Jaco). If you roll it all the way to the neck, you have a sound that is not all that dissimilar from a P-Bass pickup. It's thick, it's a little bit chunky, but being a single coil instead of a split coil, it's still thinner, not quite as chunky, and has this, I don't quite know how to describe it, waxy sound on the high end. Some people prefer that. I don't. if you're playing by yourself there's gonna be no mistaking the J neck pickup for the P pickup, but in a band setting, it does make a fairly decent substitute, and you could probably fool some people with it. When you blend the two pickups, you get "That J-Bass sound" that is deep enough to contend with a P-Bass, but nowhere near as chunky, very smooth and bright.
With a P/J combo, you get the bridge J, so can go Jaco on it, and you have the P-Bass pickup also so that you can pull off the fat, chunk that punk/funk/rock/country/blues, even jazz have really benefited from. The tradeoff, is that when you blend the P and the J, it isn't as smooth as when you blend a J and a J, some of that chunk from the P carries over. Now, some people prefer that. I don't. You'll have to try it out for yourself. You can go for the deluxe precision bass special (which is a P-Body/J-Neck and a P/J pickup configuration) It's active though. You may find you prefer the active sound as it's brighter (even with the treble all the way down), and it's boomier. But it's also less natural sounding. What I would do if you're gonna go that route would be to have it modded so that you can switch back and forth between active and passive. My active/passive P/J spends 99% of it's time in passive P mode, so it could just as easily be a standard P-Bass for the way I use it. If you want to just go for something that's passive, you could go for the Aerodyne J-Bass of Aerodyne P-Bass. Both basses are Passive P/J combos, and ultra-light weight (which has gotta thin out the sound a little bit) The J has a super sleek J-shaped, all black body w/ no pickguard and black headstock. The P is P shaped, and blue, with a white pickguard and a blue painted headstock.
The Body of the J is slightly bigger than that of a P, and also slightly offset. It's a little more artsy-fartsy than a P-Bass, but it's not as symetrical. Someone on this thread has already mentionted the whole leaning up against an amp thing, and he's right, you're gonna have to fenagle that puppy just right to get it to lean against the amp without falling over, and if you bump it, ho ho, game over.
the Neck on the J is somewhat slimmer than the P. Not so much the thickness of the neck itself, but more the width of the fretboard at the nut. Since the nut governs the spacing of the strings at that end of the bass, the strings will be slightly closer together on the J, and slightly farther apart on the P. Pros and Cons: The closer the strings, the easier it'll be to play fast, but the harder it'll be to play neat as well. The further the strings, the easier it'll be to play neat and not get sloppy, but the harder it'll be to play fast. Frankly, they're not all that much different, and you'll instantly acclimate yourself to whichever one you go for.
As far as Folk bassists and statistics on J-Bass and P-Bass users, I have no data, but I imagine that the smoother, cleaner, and less obtrusive J sound would be better, and remember, you'd have the neck pickup too, which is vaguely P-Bass like if you really need it, and if you wanted to have a solo, you'll have that bridge pickup too. As far as punk bassists, yeah, I think the majority of them use P-Bass, but a J-Bass in a punk band is by no means unheard of. And consider this, I'm seeing a lot more "Sting" P-Basses being used. Well a "Sting" style is really the original style of P-Bass with a Telecaster style headstock instead of the Stratocaster style that we've become accusomed to, and a totally different pickguard design. But most importantly, instead of the split coil pickup which we know and love as the P-Bass pickup, they used single coil....so...........therefore......people who are using "Sting" style P-Basses, are really using the Neck pickup in a J-Bass.......so they are really using J-Basses with only half the functionality of a traditional J-Bass.
I know that some folk bassists go for Fretless, especially if their folk is a little bit jazzy. And unless you wanna get into a Tony Franklin (which is $$$$$$), the only Fender option for a Fretless would be a J-Bass.
Ideally, If you can afford it, rather than seeing you settle with a P/J combo, I'd see you get a P-Bass -and- a J-Bass, really, the best thing would be if you did what I did, got the deluxe active P/J, and modded it for active/passive, then get two J-Basses, a 5-string active, and again, mod it for active/passive, and then lastly, get a standard Jazz Fretless.
But if you can only afford one bass, then my suggestion would be to decide whether or not you prefer a P to a Neck J, a P/J to a J/J, and perhaps most importantly, a P to a J/J. I have a strong hunch that you'll prefer the split coil P to the single coil J Neck, but would not be at all surprised if you prefer the sound of a J/J to a P/J. Unfortunately, if this is the case, then it puts you at an impasse. You want the traditional P sound, and are not exactly thrilled with the J Neck compromise, and you want the tradional "J" sound, and the P/J combo is just a little brutish for what you're after, then what you have to do at that point is decide which sound you like better, the P or the J/J. If the J/J, get a J-Bass and just live with the Neck pickup. It'll do ya fine. But if you like the P better, then my vote would be to either get the Deluxe Active P/J, then mod it for active/passive, or, if you don't feel like modding your bass, get the Aerodyne that fits your cosmetic sensibilties best (both will be passive P/J) and then get an external preamp so you can switch between active and passive externally. (Remember, that if you do the Aerodyne J, you'll have a harder time leaning it against things, but it is super super slick looking.)
You might also check out an Epiphone Thunderbird, or an OLP MM2/MM3 if you want passive, or MM22/MM32 if you want active (4 string/5 string) If you can afford the Gibson and Ernie Ball Music Man versions of these basses, knock yourself out, but my recommendation, especially if you're riding the fence on several models is with the money you'd spend on a single Gibson or EBMM, or a single MIA (Made in America) Fender, you could get anywhere between 2-4 of these budget versions (including MIM Fender, Made in Mexico) which, really are 80-90% of the quality of the high dollar version and cosmetically identical. Plus they are all officially licensed, and in the case of MIM Fender and Epiphone actually made by the high company. Some would argue that if you're gonna do it, pick the one you really want and then don't waste your time with a knock-off, but go for the real thing. A valid point, I suppose, but I see it as more advantageous and better financial stewardship, if you're gonna spend that kind of money, to have a broader portfolio of basses. Then your decision becomes ----a lot---- easier!
But, if you are gonna do the one high-dollar bass approach, then undoubtedly, go for the MIA Jazz Bass with S1 switching. You see, typically on a J-Bass, when you're running both pickups, you're running them in parrallel (two separate pickups running side by side), but when you activate the S1 switch instead of running them in parrallel, you're now running them in series (the two pickups become one mega pickup), and at that point, you have a dual coil pickup...if that sounds familiar, it's because the P-Bass split coil is a dual coil pickup. So, that's right, S1 off, it's a J-Bass, S1 on, then for all practical purposes, it's a P-Bass. Therefore, if you're gonna go for a high-dollar bass, this is the no-brainer, cause you get both of these legendary axes in one!
As for me personally, I prefer the sound of the split coil P to the single coil J neck hands down. But I prefer the sound of the J/J, to the P/J. They're very similar, you can tell their sister sounds, but the very same thick chunk that wins the P over the J Neck causes the P/J to be a little rougher than I want in that kind of sound. So, there, I'm stuck in the middle too. I had to decide and settle the matter of whether I preferred the P or the J/J. After a lot of thought and hours upon hours of fooling around with the same set of bass runs over and over again switching back and forth between the two basses that I have to say that by the narrowest of margins, I prefer the J/J. So my vote, if we're sticking with Fenders, goes to the MIM Fender Jazz Bass!
As long as my post already is (it should be in chapter and verse format, I think)

I wont really get into the virtues, vices and attributes of the Epiphone Thunderbird and/or OLP basses in this post, but if you're interested, either PM me, or else mention it in one of your posts on the thread and I'll break it down for you.
I hope that helps!
God Bless!!!